Question

At the start of the current week there are 30 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours and on machine B is forecast to be 35 hours. The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95 units. Company policy is to maximize the combined sum of the units of X and the units of Y in stock at the end of the week.

 Formulate the problem of deciding how much of each product to make in the current week as a linear program.

 Solve this linear program graphically.

Answer 1

A company makes two products (X and Y) using two machines (A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B.

Solution.

x is the number of units of X produced in the current week

y is the number of units of Y produced in the current week

constraints are:

$50x + 24y \le 40\cdot60$ machine A time

$30x + 33y \le 35\cdot60$ machine B time

$x \ge 75 - 30$

i.e. $x \ge 45$ so production of X$\ge$ demand (75) - initial stock (30), which ensures we meet demand

$y\ge 95 - 90$

i.e. $y\ge 5$ so production of Y $\ge$ demand (95) - initial stock (90), which ensures we meet demand

The objective is: maximise $(x+30-75) + (y+90-95) = (x+y-50)$

i.e. to maximise the number of units left in stock at the end of the week

It is plain from the diagram below that the maximum occurs at the intersection of

$x=45$ and $50x + 24y = 2400$

we have that x=45 and y=6.25 with the value of the objective function being 1.25